# Stochastic Modeling of Chemical Compounds in a Limestone Deposit by Unlocking the Complexity in Bivariate Relationships

^{*}

## Abstract

**:**

_{2}O

_{3}, Fe

_{2}O

_{3}, and SiO

_{2}) with complexity in bivariate relationships are analyzed and 100 realizations are produced for each variable. To show the effectiveness of the proposed algorithm, the outputs (realizations) are statistically examined and the results show that this methodology is legitimate for reproduction of original mean, variance, and complex cross-correlation among the variables and can be employed for further processes. Then, the applicability of the concept is demonstrated on a workflow to classify this limestone deposit as measured, indicated, or inferred based on Joint Ore Reserves Committee (JORC) code. The categorization is carried out based on two zone definitions, geological, and mining units.

## 1. Introduction

## 2. Methodology

#### 2.1. Gaussian (Co)-Simulation

#### 2.2. Projection Pursuit Multivariate Transform Steps

#### 2.2.1. Preprocessing Steps

#### Normal Score Transformation

^{−1}are original data cumulative distribution function (CDF) standard and normal data CDF, respectively.

#### Data Sphering

#### Projection Pursuit

^{T}by X(AZ), which results in back-transformation to the initial basis (Equation (6)):

#### Stopping Criteria

#### Application

#### 2.3. Proposed Algorithm

- Exploratory data analysis of multivariate data
- Investigation of the level of complexity in bivariate relation analysis
- PPMT forward transformation
- Examination of removing cross-correlations among variables by using cross-correlogram
- Inference of cross-dependency functions by linear model of co-regionalization (LMC)
- (Co)-simulation of PPMT transformed factors taking into account the fitted LMC
- PPMT backward transformation of simulated results (realizations) into original scale
- Validation of the output by statistical analysis tools

## 3. Case Study: Aktas-South Deposit in Kazakhstan

#### 3.1. Exploratory Data Analysis in Limestone Deposit

_{2}O

_{3}), aluminum oxide (Al

_{2}O

_{3}), silicon oxide (SiO

_{2}), and calcium oxide (CaO), whose values are assayed in percentages. Similar to all geostatistical projects, the first step is exploratory data analysis to identify global statistical characteristics of the underlying variables. First, possible outliers and duplicated data were recognized. The presence of outliers in the dataset makes the inference of statistical parameters problematic and nonrepresentative [37,38]. These aberrant values intentionally influence the variance and result in sharp fluctuations in variogram analysis [39]. In addition, detection and removal of duplicate data is also important prior to any geostatistical analysis. One of the problems is that these repeated values generate singular matrices in kriging systems, leading to unestimated blocks surrounding the duplicate locations [40]. After removing duplicate locations in this study, the variables are plotted in a probability plot. This statistical tool helps to detect and fix extreme and innermost values [41]. Following this, some outlier values are detected for Fe

_{2}O

_{3}and CaO (Figure 3). In the distribution of Fe

_{2}O

_{3}, values more than about 18% are considered as extreme values, and in the distribution of CaO, values less than about 19% are recognized as innermost values, while the other two distributions (SiO

_{2}and Al

_{2}O

_{3}) sound reasonable given that no significant outliers were identified.

_{2}. These two variables introduce high-quality limestone for high amounts of CaO (>10%) and low amounts of SiO

_{2}(<40%). In addition, Fe

_{2}O

_{3}plays an important role, yet does not have as significant an influence as SiO

_{2}in favor of CaO. As can be seen, the majority of drillholes illustrate a high concentration of CaO, which is distributed homogeneously in the region. Interestingly, SiO

_{2}reveals poor concentration in the same locations, meaning it satisfies the quality of limestone for the entire deposit, corroborating high CaO and low SiO

_{2}. This visual inspection also indicates a strong spatial dependency between CaO and SiO

_{2}, in which there is a negative impact on the local distribution of these two chemical compounds. This shows a high concentration of CaO versus a low concentration of SiO

_{2}, which is important for this deposit. This phenomenon motivates a further investigation into the cross-correlation structures among these chemical variables toward better decision making for the selection of efficient geostatistical algorithms for 3D modelling and mineral resource evaluation.

_{2}, is less than 2.0, which indicates that the distribution of data has no significant harsh variability and predictive models can be suitable and meaningful [41]. In this limestone deposit, as previously mentioned, CaO and SiO

_{2}are two critical variables for ore/waste selection. For this purpose, areas with more than 10% CaO and less than 40% SiO

_{2}define ore zones, and areas with less than 10% CaO and more than 40% SiO

_{2}introduce waste zones based on mining excavation destination. Before initiating the modelling process, it might be of interest to calculate two global recovery functions, fraction of recoverable ore above or below the cutoff and mean grade above or below the cutoff [28,42]. These two parameters are calculated by bivariate cumulative distribution functions computed over CaO and SiO

_{2}as follows:

_{2}O

_{3}and CaO (−0.64) and between Fe

_{2}O

_{3}and SiO

_{2}(approximately 0.53). The highest dependency is seen between CaO and SiO

_{2}, which has negative correlation with almost −0.94 correlation coefficient. This corroborates the visual interpretation already provided in the location map of sampling points (Figure 4). Other correlations, such as those between Fe

_{2}O

_{3}and Al

_{2}O

_{3}, Al

_{2}O

_{3}and CaO, and Al

_{2}O

_{3}and SiO

_{2}, are somewhat low. These values only give a general perspective on the linear dependency that exists among the variables and may not be suitable for examining whether or not complex characteristics such as nonlinearity, heteroscedasticity, and geological constraints may exist. In order to examine the latter characteristics, the bivariate relation in Figure 5 is presented as scatter plot between pairs of the variables Al

_{2}O

_{3}, CaO, Fe

_{2}O

_{3}, and SiO

_{2}. This statistical diagram is suitable to explore relationships such as complexities and linearity features between pairs of variables. This is an interesting illustration of different complexities between co-variables, starting from heteroscedastic characteristics between Fe

_{2}O

_{3}and Al

_{2}O

_{3}, and CaO and SiO

_{2}; nonlinearity observed between SiO

_{2}and CaO; and geologic constraints between CaO and Al

_{2}O

_{3}.

#### 3.2. PPMT Forward Transformation

_{2}O

_{3}, CaO, Fe

_{2}O

_{3}, and SiO

_{2}). The result of this modelling approach is then applied for mineral resource classification. However, prior to any geostatistical modelling, whether it is independent simulation or co-simulation after forward PPMT transformation, the removal of correlations after this first transformation should be assessed. Therefore, in this study, as a common practice in forward PPMT transformation, the variables are first transformed to PPMT factors and then undergo one further transformation by MAF, implemented to completely remove cross-correlation. This can be evaluated by a cross-correlogram (Figure 6). The results of the cross-correlogram between MAF factors show that a small amount of correlation is still resistant in some lags. For instance, one can recognize that the correlation in the first lag (~100 m) is around 25% between SiO

_{2}and Al

_{2}O

_{3}even after this MAF transformation (Figure 6). This signifies that MAF is not able to decorrelate the variables over all the lag separations. In this respect, it is not advocated to use the independent simulation due to the remaining small correlations among factors. In order to cope with the proposed algorithm in this case, it was decided to employ co-simulation right after the initial PPMT forward transformation of underlying variables irrespective of considering any further MAF transformation. For this, once again, the cross-correlation among the transformed PPMT factors is examined through the cross-correlogram and before the MAF step. The results show that the correlation manifests itself through some lags (e.g., ~12% between SiO

_{2}and CaO at a lag separation of 150 m and ~11% between SiO

_{2}and Al

_{2}O

_{3}at a lag separation of 400 m; Figure 7), although at a lag separation of 0, the correlation is significantly removed (Figure 8). Even these small amounts of correlation among the PPMT factors are important and provoke applying the co-simulation algorithms via PPMT transformed factors and considering the linear model of co-regionalization.

#### 3.3. Variogram Inference

#### 3.4. Stochastic Modeling in Limestone Deposit

_{2}O

_{3}, Fe

_{2}O

_{3}, and SiO

_{2}were produced through 100 realizations, as shown in Figure 10. Before further analysis over the simulated results, it is necessary to check whether the outputs of this proposed algorithm are statistically valid.

#### 3.5. Validation

_{2}O

_{3}, Al

_{2}O

_{3}, SiO

_{2}, and CaO) and the means of simulated and back-transformed variables calculated from PPMT through 100 realizations is shown in Figure 11. As can be seen, PPMT is able to reproduce the original mean values of each variable.

_{2}O

_{3}, Al

_{2}O

_{3}, SiO

_{2}, and CaO) and simulated and back-transformed values obtained from the PPMT method through 100 realizations. For CaO and SiO

_{2}(Figure 12), it is intuitively determined that PPMT produces satisfactory outputs in terms of reproduction of variance. However, minor deviations, such as for Fe

_{2}O

_{3}, as can be seen from this figure, are referred to the influence of conditioning data [24,50,51]. However, this tiny departure of average of variances for 100 realizations from original variance is not remarkably significant.

_{2}O

_{3}, Al

_{2}O

_{3}, SiO

_{2}, and CaO) through 100 realizations. As it can be seen from Figure 13, co-simulation methodology shows satisfactory results in the reproduction of original correlation coefficients among co-variables. This good performance among variables can be explained by the fact that co-simulation considers the intrinsic correlation between variables by incorporating the linear model of co-regionalization [51,52,53].

_{2}O

_{3}and Al

_{2}O

_{3}, PPMT co-simulation successfully reproduces the shape of original correlation. The same features are evidently demonstrated in the reproduction of the shape of original correlation between Fe

_{2}O

_{3}and CaO, Fe

_{2}O

_{3}and SiO

_{2}, and SiO

_{2}and CaO. However, inadequate results in the reproduction of the shape of correlation between SiO

_{2}and Al

_{2}O

_{3}, and CaO and Al

_{2}O

_{3}can be seen. Overall, the proposed approach based on the combination of PPMT and co-simulation is capable of reconstructing the original shape of correlation.

#### 3.6. Mineral Resource Classification

_{2}O

_{3}are two critical components for the underlying zone. The second is mainly concerned with mining excavation units where CaO, SiO

_{2}, and Al

_{2}O

_{3}are vital variables for this zone definition. In the following, we present the results for both types of mineral resource classification that define the different categories based on ore zone definitions and mining units.

#### 3.6.1. Ore Zone Definitions

#### 3.6.2. Mining Units

_{2}, and Fe

_{2}O

_{3}. As also shown in Table 3, there is strong correlation among these three variables. The results of the proposed approach, in fact, show that the correlation coefficients are reproduced at a satisfactory level of confidence. The importance of this is shown in this section; for instance, these three variables are the key factors in favor of grouping the Aktas-South limestone deposit for mining excavation, which consequently impacts the rigorous classification of mineral resources. The list of mining units based on chemical cutoffs is shown in Table 6. It should be mentioned that in this classification, the number of restrictions (chemical cutoffs) reaches four, which means that the results of classification will be according to seven restrictions. As can be seen, CaO and SiO

_{2}are two chemical compounds that define the ore/waste classification technique. The tonnage for each classification based on JORC code in this deposit is summarized in Table 7, and all results of the mentioned method with resource classification are summarized in single graphs for highly green limestone (HGLS), brown limestone (BROWNLS), ferrous limestone (FEROLS), cherty limestone (CHERTYLS), cherty limestone 2 (CHERTYLS2), MARL, and WASTE in Figure 16, following the same method as explained in the previous section [41].

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic pattern of (

**a**) multi-Gaussianity; multivariate complexities such as (

**b**) constraint, (

**c**) non-linearity and (

**d**) heteroscedasticity.

**Figure 5.**Scatter plots of (

**a**) Fe

_{2}O

_{3}and Al

_{2}O

_{3}, (

**b**) Fe

_{2}O

_{3}and CaO, (

**c**) Fe

_{2}O

_{3}and SiO

_{2}, (

**d**) SiO

_{2}and Al

_{2}O

_{3}, (

**e**) SiO

_{2}and CaO, (

**f**) CaO and Al

_{2}O

_{3}.

**Figure 6.**Correlograms of PPMT factors with minimum/maximum autocorrelation factor (MAF) transformation ((

**a**) PPMT Fe

_{2}O

_{3}and PPMT Al

_{2}O

_{3}, (

**b**) PPMT Fe

_{2}O

_{3}and PPMT CaO, (

**c**) PPMT SiO

_{2}and PPMT Fe

_{2}O

_{3}, (

**d**) PPMT SiO

_{2}and PPMT Al

_{2}O

_{3}, (

**e**) PPMT SiO

_{2}and (

**f**) PPMT CaO, PPMT CaO and PPMT Al

_{2}O

_{3}).

**Figure 7.**Correlograms of PPMT factors without MAF transformation ((

**a**) PPMT Fe

_{2}O

_{3}and PPMT Al

_{2}O

_{3}, (

**b**) PPMT Fe

_{2}O

_{3}and PPMT CaO, (

**c**) PPMT SiO

_{2}and PPMT Fe

_{2}O

_{3}, (

**d**) PPMT SiO

_{2}and PPMT Al

_{2}O

_{3}, (

**e**) PPMT SiO

_{2}and PPMT CaO, (

**f**) PPMT CaO and PPMT Al

_{2}O

_{3}).

**Figure 8.**Scatter plots of transformed variables without integration of MAF: (

**a**) PPMT_Al

_{2}O

_{3}and PPMT_Fe

_{2}O

_{3}, (

**b**) PPMT_Al

_{2}O

_{3}and PPMT_SiO

_{2}, (

**c**) PPMT_CaO and PPMT_Fe

_{2}O

_{3}, (

**d**) PPMT_CaO and PPMT_SiO

_{2}, (

**e**) PPMT_Al

_{2}O

_{3}and PPMT_CaO, (

**f**) PPMT_Fe

_{2}O

_{3}and PPMT_SiO

_{2}. Correlation at lag 0 is almost zero.

**Figure 9.**Fitted direct variograms of transformed variables: (

**a**) PPMT Al

_{2}O

_{3}, (

**b**) PPMT CaO, (

**c**) PPMT SiO

_{2}, (

**d**) PPMT SiO

_{2}. For brevity, only direct variograms are presented.

**Figure 11.**Histograms of mean values of: (

**a**) aluminium oxide (Al

_{2}O

_{3}), (

**b**) calcium oxide (CaO), (

**c**) iron oxide (Fe

_{2}O

_{3}) and (

**d**) silicon oxide (SiO

_{2}) obtained from PPMT method. Green line is the average of mean values over 100 realizations; red line represents the original mean of variables.

**Figure 12.**Histograms of mean variance of: (

**a**) aluminium oxide (Al

_{2}O

_{3}), (

**b**) calcium oxide (CaO), (

**c**) iron oxide (Fe

_{2}O

_{3}), and (

**d**) silicon oxide (SiO

_{2}) obtained from PPMT method. Green line is the average mean of 100 realizations; red line represents the original mean of variables.

**Figure 13.**Graphs of correlation coefficients of: (

**a**) Al

_{2}O

_{3}and Fe

_{2}O

_{3}, (

**b**) Al

_{2}O

_{3}and CaO, (

**c**) Al

_{2}O

_{3}and SiO

_{2}, (

**d**) CaO and Fe

_{2}O

_{3}, (

**e**) CaO and SiO

_{2}, (

**f**) and Fe

_{2}O

_{3}and SiO

_{2}obtained by PPMT method. Green line is the average mean of 100 realizations; red line represents the original correlation coefficient between variables.

**Figure 14.**Reproduction of original correlation coefficient (green) between (

**a**) Fe

_{2}O

_{3}and Al

_{2}O

_{3}, (

**b**) Fe

_{2}O

_{3}and CaO, (

**c**) Fe

_{2}O

_{3}and SiO

_{2}, (

**d**) SiO

_{2}and Al

_{2}O

_{3}, (

**e**) SiO

_{2}and CaO, (

**f**) CaO and Al

_{2}O

_{3}by PPMT method (red).

**Figure 15.**Resource classification of CaO in different ore zones: 1: marl-chert; 2: pale yellow limestone; 3: brown limestone.

**Figure 16.**Resource classification of CaO in different mining units: 1: HGLS; 2: BROWNLS; 3: FEROLS; 4: CHERTYLS; 5: CHERTYLS2; 6: MARL; 7: WASTE.

Lithology | Physical Appearance | Chemical Characteristics | Comment |
---|---|---|---|

Cherty limestone (CL) | Yellow in color with alternating cherty bands | Not ascertained | Likely to be used in cement after removing cherty bands |

Pale yellow limestone (PYLS) | Yellow to pale brown in color | >40% CaO and <3.5% Fe_{2}O_{3} | Very good for cement manufacturing |

Brown cherty limestone (BCLS) | Brown to dark brown | >8% Fe_{2}O_{3} | |

Ferruginous limestone (FLS) | Brown to dark brown | >40% CaO and Fe_{2}O_{3} > 3.5% to 10 | Considered as low grade limestone |

**Table 2.**Statistical univariate parameters of original Al

_{2}O

_{3}, CaO, Fe

_{2}O

_{3}, and SiO

_{2}in the dataset of Aktas-South deposit.

Variable (%) | Number of Samples | Minimum | Maximum | Mean | Variance | Coefficient of Variation (COV) |
---|---|---|---|---|---|---|

Al_{2}O_{3} | 4553 | 0.21 | 42.32 | 9.33 | 132.92 | 1.23 |

CaO | 4553 | 0.75 | 53.94 | 38.79 | 164.14 | 0.33 |

Fe_{2}O_{3} | 4553 | 0.26 | 38.24 | 4.29 | 9.04 | 0.70 |

SiO_{2} | 4553 | 0.03 | 99.37 | 27.24 | 744.72 | 1.00 |

Variables | Fe_{2}O_{3} | Al_{2}O_{3} | CaO | SiO_{2} |
---|---|---|---|---|

Fe_{2}O_{3} | 1 | 0.13 | −0.64 | 0.53 |

Al_{2}O_{3} | 1 | −0.17 | 0.13 | |

CaO | 1 | −0.94 | ||

SiO_{2} | 1 |

Zone | Chemical Cutoffs |
---|---|

Marl-chert | ≤40% and ≥20% of CaO |

Pale yellow limestone | >40% of CaO and <3% of Fe_{2}O_{3} |

Brown limestone | >40% of CaO and ≥3% and <4.5% of Fe_{2}O_{3} |

Classification | Marl-Chert (Mt) | Pale Yellow Limestone (Mt) | Brown Limestone (Mt) |
---|---|---|---|

Measured | 53 | 102.14288 | 52.56608 |

Indicated | 30 | 10.79 | 5.4357 |

Inferred | 125 | 67.99632 | 33.652 |

Total | 208 | 180.9292 | 91.65378 |

Mining Unit | Chemical Cutoffs |
---|---|

HGLS | CaO ≥ 40 and SiO_{2} ≤15 and Fe_{2}O_{3} < 3 |

BROWNLS | CaO ≥ 40 and SiO_{2} ≤ 15 and Fe_{2}O_{3} ≥ 3 and Fe_{2}O_{3} < 4 |

FEROLS | CaO ≥ 40 and SiO_{2} ≤ 15 and Fe_{2}O_{3} ≥ 4 |

CHERTYLS | CaO < 50 and CaO > 20 and SiO_{2} > 15 and SiO_{2} ≤40 |

CHERTYLS2 | CaO < 40 and CaO > 30 and SiO_{2} ≤15 |

MARL | CaO < 45 and CaO > 10 and SiO_{2} > 40 |

WASTE | CaO ≤ 10 and CaO > 0 and SiO_{2} > 40 |

Category | HGLS | BROWNLS | FEROLS | CHERTYLS | CHERTYLS2 | MARL | WASTE |
---|---|---|---|---|---|---|---|

Measured tonnage | 95.8 | 38.6 | 29.3 | 14.9 | 0.4 | 53.4 | 0.0 |

Indicated tonnage | 9.3 | 2.8 | 3.4 | 13.5 | 1.5 | 20.4 | 0.0 |

Inferred tonnage | 63.4 | 24.1 | 18.9 | 26.8 | 1.8 | 113.8 | 1.2 |

Total tonnage | 168.5 | 65.6 | 51.5 | 55.1 | 3.7 | 187.5 | 1.2 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Battalgazy, N.; Madani, N.
Stochastic Modeling of Chemical Compounds in a Limestone Deposit by Unlocking the Complexity in Bivariate Relationships. *Minerals* **2019**, *9*, 683.
https://doi.org/10.3390/min9110683

**AMA Style**

Battalgazy N, Madani N.
Stochastic Modeling of Chemical Compounds in a Limestone Deposit by Unlocking the Complexity in Bivariate Relationships. *Minerals*. 2019; 9(11):683.
https://doi.org/10.3390/min9110683

**Chicago/Turabian Style**

Battalgazy, Nurassyl, and Nasser Madani.
2019. "Stochastic Modeling of Chemical Compounds in a Limestone Deposit by Unlocking the Complexity in Bivariate Relationships" *Minerals* 9, no. 11: 683.
https://doi.org/10.3390/min9110683